ON THE EXISTENCE OF MINIMAL β-POWERS

Abstract

If all proper factors of a word u are β-power-free while u itself is not, then u is a minimal β-power. We consider the following general problem: for which numbers k,β, and p there exists a k-ary minimal β-power of period p? For the case β ≥ 2 we completely solve this problem. If the number β < 2 is relatively "big" w.r.t. k, we show that any number p can be the period of a minimal β-power. Finally, for "small" β we describe some sets of forbidden periods and provide a numerical evidence that for k ≥ 9 these sets are almost exhaustive.